Properties

Label 2-40560-1.1-c1-0-29
Degree $2$
Conductor $40560$
Sign $1$
Analytic cond. $323.873$
Root an. cond. $17.9964$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2·7-s + 9-s + 4·11-s + 15-s + 8·17-s − 6·19-s + 2·21-s − 6·23-s + 25-s + 27-s − 4·29-s + 4·33-s + 2·35-s + 2·37-s + 2·41-s + 4·43-s + 45-s − 3·49-s + 8·51-s − 10·53-s + 4·55-s − 6·57-s + 4·59-s − 10·61-s + 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.258·15-s + 1.94·17-s − 1.37·19-s + 0.436·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.696·33-s + 0.338·35-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.149·45-s − 3/7·49-s + 1.12·51-s − 1.37·53-s + 0.539·55-s − 0.794·57-s + 0.520·59-s − 1.28·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(323.873\)
Root analytic conductor: \(17.9964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{40560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 40560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.394265749\)
\(L(\frac12)\) \(\approx\) \(4.394265749\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.63464293393849, −14.32130401618161, −13.98693800458677, −13.26413347153166, −12.63047903751046, −12.25047540186648, −11.69404059097995, −11.10783326316673, −10.48282902240343, −9.991817564675496, −9.374650965340985, −9.073595965545615, −8.219691315708241, −7.945715878165426, −7.414638008344397, −6.528281981742826, −6.139711296228554, −5.515651803252598, −4.775099851397688, −4.109514671840894, −3.654371666366626, −2.905598884656308, −1.927060001164769, −1.673090422802711, −0.7513015752209076, 0.7513015752209076, 1.673090422802711, 1.927060001164769, 2.905598884656308, 3.654371666366626, 4.109514671840894, 4.775099851397688, 5.515651803252598, 6.139711296228554, 6.528281981742826, 7.414638008344397, 7.945715878165426, 8.219691315708241, 9.073595965545615, 9.374650965340985, 9.991817564675496, 10.48282902240343, 11.10783326316673, 11.69404059097995, 12.25047540186648, 12.63047903751046, 13.26413347153166, 13.98693800458677, 14.32130401618161, 14.63464293393849

Graph of the $Z$-function along the critical line